You're right... and I also learned yesterday about pull-down and pull-up resistors, so I'm now looking for two resistors with about 1kOhm to connect the two inputs of the buffer/line driver to GND, and then see if the circuit gets more stable. If that works fine, I'm going to write the relevant data bits directly to the serial interface, as it has a bandwidth of about 12Mbaud and thus should be able to transmit the data, so I can do the CPU-heavy work directly on the CPU.